The power of induction.

Question

I was thinking about the triangle inequality for complex values, and was wondering if this would be a sufficient proof. In attempts to show the triangle inequality for complex numbers, that is $$|z+w|\leq |z|+|w|$$ for $z$, $w \in \mathbb{C}$, would it be sufficient (allowed) to show first that $$|z_1+...+z_n|\leq |z_1|+...+|z_n|.$$ Proving this by induction, $n=1$ we have that $$|z_1|\leq|z_1|$$ which is trivially true, and then jump into the inductive step and use suppose that $$|z_1+...+z_k|\leq |z_1|+...+|z_k|$$ is true, and then show that $$|z_1+...+z_{k+1}|\leq |z_1|+...+|z_{k+1}|$$ is true. And then one lets $n=2$ and claim proof of the triangle inequality. This seems that it is skipping the crux of the triangle inequality in that it is really only interesting with at least two values, but induction seems to suppose otherwise. So my question is, "is induction really that powerful?" or am I miss understanding what induction does?

Answer 1

This reminds me of the fake inductive proof that "all horses are the same color":

1. One horse is the same color.
2. Given $n+1$ horses, the first $n$ are the same color by inductive hypothesis, as are the last $n$. Let $h$ be a horse belonging to both subsets. So all $n+1$ horses are the same color as $h$.

When you say "jump into the inductive step", you don't say how you show that the triangle inequality for $n$ numbers implies it for $n+1$ numbers. That's the crux of the argument. If you write it out, you'll see it relies on the triangle inequality for two numbers.

To answer your question, what is induction good for: OK, let's say we've proven the triangle inequality for two numbers by another method. (The standard proofs don't use induction.) It's still useful to know that it holds for any set of $n$ numbers.

Now, you could write an informal argument, bootstrapping from 2 to 3, from 3 to 4, and then the pattern would be clear; you could then say, "etc.". But to make this argument rigorous and formal, you need induction,

As to why you might want a rigorous and formal argument, that's another topic.